DB versus DC: A Comparison of TotalCompensation

by

Yueren Wang

B.Sc., Simon Fraser University, 2013

Project Submitted in Partial Fulfillment of theRequirements for the Degree of

Master of Science (Actuarial Science)

in theDepartment of Actuarial Science and Statistics

Faculty of Science

c©Yueren Wang 2018SIMON FRASER UNIVERSITY

Summer 2018

Copyright in this work rests with the author. Please ensure that any reproductionor re-use is done in accordance with the relevant national copyright legislation.

Approval

Name: Yueren Wang

Degree: Master of Science (Actuarial Science)

Title: DB versus DC: A Comparison of TotalCompensation

Examining Committee: Chair: Yi LuProfessor

Gary ParkerSenior SupervisorAssociate Professor

Barbara SandersSupervisorAssociate Professor

Carl SchwarzInternal ExaminerProfessor

Date Defended: July 6, 2018

ii

Abstract

Employer-sponsored pension plans play an important role in providing employees with ade-quate retirement income. They are expensive and carry some important risks. The employerand its employees share these costs and risks differently depending on the plan design. Inthis project, two designs are studied, a defined benefit (DB) plan and a defined contri-bution (DC) plan. They are analyzed in a simple common business setup under the samestochastic economic scenarios generated from a calibrated VAR model. The employer’s totalcompensation budget is assumed to be constrained so that higher pension contributions areassociated with lower salary increases, and vice versa. The two types of plans are comparedbased on the total compensation, defined as the value of wages and retirement income,received by 25 cohorts of new employees. On an adjusted basis, we find that the two typesof plans provide equivalent total compensation to their members.

Keywords: Pension plan; DB plan; DC plan; salary; contribution; retirement income; totalcompensation

iii

Dedication

To my beloved Laoba, Laoma, mother, husband and my adorable daughter!

iv

Acknowledgements

First and foremost, I would like to express my grateful appreciation to my senior su-pervisor, Dr. Gary Parker. His generous help, continuous understanding, and constructiveinspiration contributed to the successful completion of this project. It is him who encour-aged me when I found it difficult being a graduate student and taking care of my baby girlsimultaneously throughout my master’s studies. Beyond the knowledge, it is the way of howto conduct a research study that he really taught me and I benefited a lot from that. Hisrigorous scholarship and responsible attitude will be my guidance for my future career. Itis absolutely my honor to be supervised by him.

I wish to thank my supervisor Ms. Barbara Sanders for her guidance and for taking timefrom her busy schedule to review my thesis. I would also like to thank Dr. Carl Schwarzfor agreeing to be my examining committee member. His valuable comments helped clarifyand improve the thesis.

I owe my gratitude to the Department of Statistics and Actuarial for its support, inparticular to the professors whose lectures I attended. I thank my friends and fellow graduatestudents for their help and encouragement. A special appreciation goes to my mother andaunt for taking care of me and my baby daughter during my master’s studies.

Last but not least, many thanks to my husband for his silent support and endless love.

v

Table of Contents

Approval ii

Abstract iii

Dedication iv

Acknowledgements v

Table of Contents vi

List of Tables viii

List of Figures ix

1 Introduction 11.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Literature Review 4

3 Economic Scenario Generator 83.1 The VAR Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Economic Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Historical Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.5 Simulated Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Defined Benefit Plan 194.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Investment risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3 Representative DB plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.3.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3.3 Salaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

vi

4.3.4 Benefits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.4 Actuarial Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.4.1 Valuation Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 244.4.2 Actuarial Liabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.4.3 Normal Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.4.4 Pension Fund . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.4.5 Unfunded Actuarial Liability . . . . . . . . . . . . . . . . . . . . . . 28

4.5 Business Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.5.1 Expenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.5.2 Salary Increase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.6 A Deterministic Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.7 Projected Salary Increases and Contribution Rates . . . . . . . . . . . . . . 37

5 Defined Contribution plan 395.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 Risks with a DC Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.2.1 Investment Risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.2.2 Interest Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.2.3 Other Risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.3 Representative DC Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.3.1 SFU Pension Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.3.2 Design and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.4 Business Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.5 A Deterministic Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6 Comparison of DB and DC Plans 466.1 Comparison of Starting Salaries . . . . . . . . . . . . . . . . . . . . . . . . . 486.2 Comparison of Retirement Benefits . . . . . . . . . . . . . . . . . . . . . . . 496.3 Comparison of Total Salaries . . . . . . . . . . . . . . . . . . . . . . . . . . 506.4 Comparison of Total Compensation . . . . . . . . . . . . . . . . . . . . . . . 526.5 Comparison of Total Compensation Relative to Baseline . . . . . . . . . . . 54

7 Conclusion 57

Bibliography 59

Appendix A Mortality table used for retirees 62

vii

List of Tables

Table 3.1 Summary statistics and estimated VAR model . . . . . . . . . . . . . 13Table 3.2 Calibrated Variance-Covariance Matrix and its Cholesky Decomposition 14

Table 4.1 Business setup assumed at time 0, DB plan under a deterministic scenario 34Table 4.2 Information known after valuation at time 3, DB plan under a deter-

ministic scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Table 4.3 Projections used to determine salary increase, ns, at time 3, DB plan

under a deterministic scenario . . . . . . . . . . . . . . . . . . . . . . 36

Table 5.1 Individual Accounts for first 2 cohorts of employees in the DC planunder a deterministic scenario . . . . . . . . . . . . . . . . . . . . . . 45

viii

List of Figures

Figure 3.1 Historical data (annualized rates) . . . . . . . . . . . . . . . . . . . 12Figure 3.2 Average of 10,000 annual rates generated by the calibrated model . 17Figure 3.3 Standard deviation of 10,000 annual rates generated by the cali-

brated model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Figure 4.1 Projected Salary Increases for Sixty Years . . . . . . . . . . . . . . 37Figure 4.2 Projected Normal Cost Rates for Sixty Years . . . . . . . . . . . . . 38

Figure 6.1 Difference in starting salaries by cohort between DB and DC plans 48Figure 6.2 Difference in values of retirement benefits by cohort between DB and

DC plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Figure 6.3 Difference in accumulated values of salaries at 1.8% by cohort be-

tween DB and DC plans . . . . . . . . . . . . . . . . . . . . . . . . 51Figure 6.4 Difference in accumulated values of salaries at 6% by cohort between

DB and DC plans . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Figure 6.5 Difference in values of total compensation by cohort between DB and

DC plans, 1.8% case . . . . . . . . . . . . . . . . . . . . . . . . . . 53Figure 6.6 Difference in values of total compensation by cohort between DB and

DC plans, 6% case . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Figure 6.7 Baseline-adjusted difference in values of total compensation by co-

hort between DB and DC plans, 1.8% case . . . . . . . . . . . . . 55Figure 6.8 Baseline-adjusted difference in values of total compensation by co-

hort between DB and DC plans, 6% case . . . . . . . . . . . . . . 56

Figure A.1 CPM2014 Public Sector Mortality Table . . . . . . . . . . . . . . . 62

ix

Chapter 1

Introduction

1.1 Background and Motivation

During the past decades, comparisons between defined benefit (DB) pension plans anddefined contribution (DC) plans has been a hot topic in the literature. Over the last 30years, a gradual shift away from DB plans to DC plans occurred in several countries, espe-cially in the United Kingdom and the United States (Broadbent et al. 2006), and to someextent in Canada. For instance, 41% of Canadian employees were covered by a DB plan in1991, and that proportion was down to 30% in 2006. The main factors contributing to thisshift include the low portability of DB plans; a changing investment climate producing lowerstock returns and falling interest rates since 2000 which put a huge pressure on employerssponsoring defined benefit plans. Also, current generations are reluctant to contribute morein order to eliminate funding deficits generated by previous cohorts who did not save enoughto support the cost of their retirement.

By transferring the investment risk to the employees, DC plans have lower volatility ofcosts. This reduction in the risk to the employer comes at the expense of a greater risk tothe employees upon retirement. With a DC plan, retirees bear some significant risks. Theymight not have saved enough during their working life, resulting in low income during retire-ment. If they consider buying an annuity, they may have to buy it at a time when annuitiesare expensive (e.g. when interest rates are low). If they choose not to annuitize, they face therisk of living longer than their retirement account was designed for, this is the longevity risk.

Although DC plans can outperform DB plans in periods of economic growth where mar-kets are buoyant, they don’t perform as well during more turbulent periods like the financialcrisis of 2008. A recent survey by Willis-Towers-Watson showed that employees with DCplans had to work longer into their retirement years in order to support their lifestyle. Em-ployers may have to decide whether to keep employees at a declining productivity and lower

1

engagement or pay out a severance in order to terminate long-term employees. As a result,a shift from a DB plan to a DC plan does not fully transfer the risk from the employer tothe employees.

Some employers that have converted from a DB plan to DC plan are considering con-verting back to a DB plan. Brown and McInnes (2014) mention two U.S. employers in thepublic sector, the States of Nebraska and West Virginia, that had converted to DC con-verted back to DB, at least partially, because of concerns over the level of income to retirees.

Many studies compare pension plans, see, for example, the bibliography cited in Brownand McInnes (2014) and Sanders (2016). DB and DC plans have been compared in termsof the risks involved and who bears those risks, the costs (contribution rates) and theirvolatility, the level of retirement income provided by each type of plan. The comparisonis even affected by the type of employer. Private sector employers have different concernsthan public sector employers when it comes to sponsoring pension plans. Therefore, theperceived advantages of each type of plan in one sector may not apply in the other sector.

Retirement benefits are an important part of negotiated compensation packages. Al-though their main goal is to secure adequate retirement income at a reasonable cost, theycan help attract and retain qualified employees. To Quote Brown and McInnes (2014), “Re-tirement is expensive and someone has to cover the costs”. We add that securing retirementincome is a risky business and someone must bear those risks.

The main goal of this project is to compare the total compensation, that is, salaries andretirement benefits, received by members of each type of pension plans, DB and DC. Thecomparison is done in an arbitrarily chosen business setup in which an employer has a fixedbudget, indexed to inflation, to spend on salaries and pension plan costs.

1.2 Outline

The setup of the project is as follows. Chapter 2 provides some background on pensionplans focusing on studies comparing DB and DC plans. The VAR model and a descriptionof the economic scenario generations are introduced in Chapter 3. Chapter 4 describes thedesign of the defined benefit (DB) plan studied here and its funding rules in our budget-controlled environment. Simulations are used to project contributions made to the plan andsalaries earned during the working lives of 25 cohorts of new entrants. Chapter 5 comparesthe salaries earned and the lump sum retirement funds that cohorts of employees in a DCplan would get under the same economic scenarios. In Chapter 6, present values are adopted

2

as measures to compare the total compensation received by different cohorts of employeesbetween DB plan and DC plan. Chapter 7 summarizes our findings and briefly discussespossible extensions for future work.

3

Chapter 2

Literature Review

Samwick and Skinner (1998) compared actual retirement benefits provided by DB planswith those provided by DC plans. Using simulated earnings histories, they found that ran-domly selected DC plans from 1995 outperformed DB plans from 1983, by providing sub-stantially higher average and median benefits. The DB plans were selected at a time whenthey covered most workers with a pension plan. DC plans were selected shortly after theycovered more workers than DB plans. They claim that their result is robust to changes inequity returns, productivity and earnings uncertainty. However, they did find evidence thatDC plans are riskier than DB plans in the sense that there is more variability in the levelof retirement benefits. DC plans also provided worse benefits in the lower 20% of the cases.

There are limitations of their study, some discussed by the authors, that could affectthe conclusion. First, one can argue that averages and medians are not good measures tocompare an individual’s pension benefits under different pension plans. As proposed byBlake, Cairns and Dowd (2001), Value-at-Risk (VaR) measures which are percentages ofplan members who will receive more or less than a certain benefit is much more meaningfulthan the mean or median. Second, this research was done two decades ago when stocksand bonds provided relatively higher rates of returns. These results may not apply in thecurrent economic conditions of low interest rates and improved longevity of retirees.

One inspiring idea from this paper is that they pointed out that what matters is thetotal compensation package of the workers including wages and all benefits received fromthe employer. Studying this total compensation question is the main goal of this project.

Blake, Cairns and Dowd (2001) used simulations to study the ratio of DC pension toDB pension for several asset models and a range of asset allocation strategies. They foundthat DC plans can be exposed to extreme risk compared to a DB benchmark. However,the DC plan possessed the advantage of offering greater portability than a DB plan byallowing members to easily transfer their pension fund when changing jobs. Value-at-risk

4

is adopted as a robust and well-established risk measure to compare pension ratios. Unlikeestimates of the moments, VaR statistics are not driven by extreme outcomes. They foundthat VaR estimates were most sensitive to the choice of asset allocation strategy, less tothe choice of asset return parameters and least to asset return models. Another of theirconclusions is that a static asset-allocation strategy with high proportion of equity in theinvestment portfolios delivered better results than any of the dynamic strategies includ-ing lifestyle strategies. This is an important finding given that lifestyle strategies were thecornerstone of many DC plans. Nevertheless, they admitted that this might be a tentativeconclusion and further investigation is needed since with limited amount of historical data,all asset models adopted in this paper produced high mean equity returns. If the datasetor parameter values of the asset models were changed, the results may be different to someextent.

Blake, Cairns and Dowd (2003) compared different distribution programmes of retire-ment income with a conventional life annuity. Some countries have no mandatory regulationsforcing pensioners to purchase life annuities, or only impose retirees to purchase annuitiesat a specific age. Only few pensioners are voluntarily willing to annuitize their funds. Strongbequest motive, self-perceived poor health condition and willingness to manage the assetthemselves for sophisticated individuals may be key reasons for those reluctant pension-ers. As a result, alternative decumulation strategies exist such as Equity-linked annuities(ELA) and Equity-linked income-drawdown annuities with bequest. These two decumula-tion strategies together with a conventional life annuity are compared in terms of expecteddiscounted utilities, which are determined by the individuals’ levels of risk aversion. Theauthors determined the optimal asset allocations by maximizing the expected discountedutility function. The main conclusion drawn is that the optimal choice of distribution pro-gramme is determined mostly by the proportion of assets invested in equity, and is totallyinsensitive to the plan member’s bequest motive.

McCarthy (2003) adopted a lifecycle model to study the performance of a defined benefitplan bearing the investment and wage risks, and examined the tradeoffs between definedcontribution plans and defined benefit plans. The author emphasized wage risk and mod-eled individuals’ wage instead of projecting wages by fixed formulas. Since the retirementbenefit of a DB plan is determined by final salaries of the member, it makes DB plansmuch more sensitive to wage variability than DC plans. Under the assumptions made inthe paper, the author concludes that DC plans, with the flexibility of investing assets instock market, would be preferred by young worker. Since most of young workers’ wealth ishuman capital, equities provide opportunities to diversify away from human capital. As aworker ages, more is known about the final wage which lowers the uncertainty regarding thebenefit provided by DB plans. As a result, older workers would benefit from choosing DB

5

plans since it would provide a valuable diversification of wealth out of financial markets.Other interesting findings include the fact that DB plans were more valuable to workerswhen wage variability was low, equity return was low and for those individuals who werefinancially wealthier.

Over the past two decades, with the emergence of Hybrid plans, studies have startedto explore the costs, risks and intergenerational risk sharing among traditional plans andmodern plans.

Bovenberg et al. (2007) compared costs and benefits of collective pension schemes ver-sus individual schemes in terms of saving and investing behaviors over the life cycle. Theyfound that an advantage of collective pension schemes is that they can relieve borrowingconstraints and allow intergenerational risk sharing. However, this ignored the diverse spe-cific preferences and circumstances of different individuals by imposing uniform rules onheterogeneous participants and introducing other constraints. They also use a simple set-ting in which wage income is riskless.

Cui et al. (2011) studied welfare aspects of intergenerational risk sharing by conductingcomparisons among collective plans and the optimal individual DC pension scheme. Theystudy the life cycle consumption that optimizes a discounted utility function for specific pen-sion schemes. Their model assumes that employees earn a flat real labor income throughouttheir working careers. The paper concludes that collective pension schemes with IRS wouldbe welfare improved over the optimal individual benchmark, and that welfare gains for anew entrant did not come at the cost of other cohorts. In short, the schemes with IRS werezero-sum games in market value terms but were positive-sum games in welfare terms. Asa result, the recognition of the welfare aspects played an important role. They also agreedwith former papers by Campbell and Nosbusch (2007) and Gollier (2008) that schemes withIRS would be more aggressive due to their enhanced capacity of risk-bearing.

Blommestein et al. (2009) analyzed the trade-off between volatility in contributions anduncertainty of benefits from the perspective of a plan member for a variety of pension plans.It included a traditional DC plan as a benchmark and other types of pension arrangements.They adopted the funding ratio and the replacement rate as variables to indicate the levelof contributions required and the standard of living provided by benefit payments respec-tively. The arrangements with higher stability in funding ratio led to a greater variability inreplacement ratio, in other words, members who had greater security of consumption levelwhile active would face volatile consumption changes around retirement age. The simula-tion results showed that hybrid plans had higher efficiency and sustainability in risk sharingwhich ensures a more predictable pension benefit while maintaining some security in contri-

6

bution costs. The authors do mention that although it is the employer’s contribution ratethat tend to fluctuate for Defined Benefit plans, implicitly employees are also paying whencontributions increase if the total compensation is fixed. However, they do not evaluate thepension designs based on the total compensation received by the employees.

Hoevenaars and Ponds (2008) demonstrated that value-based generational accounting isan important and useful decision making tool when analyzing pension funds with intergen-erational risk sharing. In order to avoid the problem of valuation in an incomplete market,they assumed that the real wage growth is zero. Results revealed the zero-sum feature invalue terms, which implies that any policy change will inevitably lead to value transfersamong generations of participants. For illustration, a less risky asset mix or a reallocationof risk bearing from flexible benefits to flexible contributions would benefit older membersat the expense of younger members.

7

Chapter 3

Economic Scenario Generator

3.1 The VAR Model

The vector autoregressive (VAR) model is a flexible and easily implemented stochasticprocess that captures the linear interdependencies between multiple time series. It explainsthe evolution of each variable as a linear function of its past value as well as the past valuesof the other variables in the model, and random innovations.

As in Li (2017), we adopt a first-order VAR process to model our selection of economicvariables. The m-dimensional VAR process can be written as:

zt+1 = Φzt + Pεt+1, (3.1)

where zt is a (m × 1) vector consisting of mean-adjusted variables and εt+1i.i.d∼ N(0, I) is a

(m × 1) vector of innovations. Φ and P are both (m × m) matrices. Φ is the correlation co-efficient matrix of the VAR model. P is the Cholesky decomposition of the contemporaneousvariance-covariance matrix Σ of the innovations. More precisely, P is a lower triangular non-singular matrix with positive diagonal elements such that the variance-covariance matrix isequal to the following product

Σ = PP T . (3.2)

Consequently,

E(Pεt(Pεt)T

)= Σ. (3.3)

The four economic variables used in this project are inflation, short-term interest rate,long-term interest rate and equity return. They are described in more detail in the nextsection. Denoting the vector of our four economic variables at time t by xt, the VAR model

8

that will be estimated from past data and used to generate future economic scenarios isgiven by Equation (3.1) where

zt = xt − µ, (3.4)

and µ is the mean vector of the process.

The model can be rewritten as

xt+1 − µ = Φ(xt − µ) + Pεt+1. (3.5)

3.2 Economic Variables

The comparisons made between DB plans and DC plans in this project rely on futureprojections of four economic variables. The salaries paid to workers and the contributionsmade to the pension plans will depend on the total budget of the employer. This budget willbe assumed to increase at a rate consistent with inflation. Assets will be allocated betweenfixed-income securities and equities. The actuarial liabilities of the plans will be valued at arate that is a function of short-term and long-term interest rates, as well as equity returns.The assumptions, plan designs and the business setup are discussed in the next two chapters.

In this section, we describe the four variables used in the VAR model. We follow thenotation introduced in Li (2017) and use a tilde to distinguish monthly rates from the cor-responding annual rates.

On account of the monetary policy adopted by the Bank and the Government of Canadain 1991, and renewed regularly since then, which aims at keeping inflation within a targetrange of 1 to 3 percent, data prior to 1991 are excluded. A dataset of rates at an annualfrequency would contain too few data points to reliably estimate the parameters of the VARmodel. At the highest frequency readily available, that is daily values, the manipulation ofthe data set would become tedious and unnecessary. Consequently, monthly data from May1991 to April 2016 are used in this project.

The percentage change in the Consumer Price Index(CPI) is a widely used indicator ofinflation. Inflation rates are calculated from monthly CPI data available in CANSIM CPITable 326-0020. Here, the instantaneous monthly rate of inflation during period (t-1, t], λt,is defined as:

9

λt = lnCPItCPIt−1

, (3.6)

where CPIt is the value of the Consumer Price Index at the end of month t. The annualizedrate of inflation is λt = 12λt.

The yield on 1-month Treasury bill is used as a proxy for the short-term interest rate.The historical data for this variable was obtained from CANSIM Table 176-0043. The yieldquoted for month t, which we denote by P 1

t , is quoted as an annual effective rate, and asa percentage. We get the instantaneous monthly short-term rate, y1

t using the followingtransformation:

y1t = 1

12 ln(

1 + P 1t

100

). (3.7)

As a proxy for the long-term interest rate, we arbitrarily choose the yield on a 10-year zero-coupon bond. These historical yields are extracted from the website of the Bankof Canada, where yields on zero-coupon bonds with maturity dates ranging from threemonths to 30 years are available on a daily basis. The 10-year annual effective yields, i120

t ,were retrieved at a monthly frequency, using values quoted on the first trading day of eachmonth. The superscript stands for 120 months until maturity. For consistency, the annualeffective yields are transformed into instantaneous monthly rates, y120

t . The monthly rateat time t is:

y120t = 1

12 ln(1 + i120

t

). (3.8)

For the equity return, we use the total return on Canadian equities which includes thecapital gains and the distribution of dividends over time. The monthly Canadian dividendyields, DYt , and the S&P/TSX composite index values, CIt, at the end of month t wereretrieved from CANSIM Table 176-0047 (where the index for year 2000 is set at 1000). Wedefine the instantaneous monthly rate of return on Canadian equities during month t as:

πSt = ln

[CItCIt−1

+ DYt12

], (3.9)

10

3.3 Historical Data

Plots of historical annualized rates for the four economic variables from 1991 to 2016 arepresented in Figure 3.1. Recall that the historical values consist of monthly rates observedat a monthly frequency. These observed monthly rates have been multiplied by 12 to getcorresponding annualized values and are therefore shown in the same scale as the one mostcommonly used in practice.

Overall, there is no obvious trend in inflation and in equity returns. Inflation rates areless volatile than equity returns. The fact that the annualized inflation rates appear to bequite volatile given Canada’s monetary policy is due to the measurement period of onemonth. The monetary policy has a target of 1 to 3 percent inflation per annum over amedium horizon. Aggregating inflation rates over a period of a few months would revealthat the monetary policy has indeed kept inflation around its target value since 1991. Largenegative equity returns in 1998 and 2008 reflect some financial crisis during those periods.As for bond yields, one notices a downward trend during the chosen period and muchsmaller fluctuations over short periods. This may suggest that bond yields exhibit somestrong auto-correlation. Finally, the yields on short-term bonds have been more volatile,and tended to be lower, than those for long-term bonds since 1991.

3.4 Estimation

The VAR model was estimated using the R package MTS. First the model was esti-mated without any constraints. The estimated parameters were obviously similar to, butnot exactly the same as, the corresponding parameters shown in Table 2.1 of Li (2017).Removing the U.S. equity data from the model produced only slightly different estimatesfor the reduced model. The estimates of the last row of the correlation coefficient matrix,Φ, suggested that the dependency of the equity return for a given month on past economicvariables (including equity itself) is not important. Therefore, equity returns can be mod-eled by a White Noise process independent of other variables. It was decided to make this afeature of the VAR model by forcing the last row of the matrix Φ to be 0 when estimatingthe model. This was done by using the V AR command in the MTS package.

Key statistics and estimated parameters of the VAR model are summarized in Table3.1. Note that since the observations were collected at a monthly frequency, the parametersare appropriate to project economic scenarios at a monthly time step.

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Figure 3.1: Historical data (annualized rates)

Annualized rates for the four economic variables used in the VAR model. The historical data was retrievedfrom CANSIM tables and the Bank of Canada website at a monthly frequency from May 1991 to April 2016.The monthly rates were calculated using (3.6)-(3.9) and multiplied by 12 to get the corresponding annualrates.

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Table 3.1: Summary statistics and estimated VAR model

a) Summary statistics λt y1t y120

t πStµ 0.00148 0.00257 0.00403 0.00658σ 0.00338 0.00171 0.00173 0.04199

b) Correlation matrix, Φ λt y1t y120

t πSt

λt+1 0.15818 0.09065 -0.09744 0.01010y1t+1 -0.00065 0.95112 0.02995 -0.00003y120t+1 0.00082 0.02354 0.97101 -0.00023πSt+1 0 0 0 0

c) Covariance matrix, Σ λt y1t y120

t πSt

λt 1.0832x10−5 9.1654x10−8 7.1008x10−8 1.5816x10−5

y1t 9.1654x10−8 7.8612x10−8 7.6624x10−9 -4.5870x10−7

y120t 7.1008x10−8 7.6624x10−9 3.6238x10−8 -5.3583x10−7

πSt 1.5816x10−5 -4.5870x10−7 -5.3582x10−7 1.7621x10−3

d) Cholesky matrix, P λt y1t y120

t πSt

λt 3.2912x10−3 0 0 0y1t 2.7848x10−5 2.7899x10−4 0 0y120t 2.1575x10−5 2.5311x10−5 1.8743x10−4 0πSt 4.8057x10−3 -2.1238x10−3 -3.1251x10−3 4.1530x10−2

Historical equity returns have the highest average and largest volatility among the fourvariables. Inflation is weakly correlated with its immediate past value. Based on the lastcolumn of panel b), inflation depends more on the past equity return than the bond yieldsdo. The second and third elements on the diagonal of Φ are fairly close to 1 suggesting, asexpected, that bond yields in consecutive months are highly correlated. Panel (c), with itssmall values in rows 2 and 3 relative to other values in Σ, confirms that bond yields arerelatively stable over short periods.

The environment in which the DB plan is assumed to operate (and described in thenext Chapter) makes the contribution rates and salary increases quite sensitive to extremereturns on equity. Further, it is known that when projecting DB plans over a long period,funds may end up so large that contribution holidays will occur over long periods, or theplan may end up seriously underfunded requiring unreasonably large contribution rates.Since the goal is to compare DB and DC plans in normal operating conditions, it is deemedappropriate to limit the number of extreme scenarios. With the estimated parameters of the

13

Table 3.2: Calibrated Variance-Covariance Matrix and its Cholesky Decomposition

c) Covariance matrix, Σ λt y1t y120

t πSt

λt 1.0832x10−5 9.1654x10−8 7.1008x10−8 7.9082x10−6

y1t 9.1654x10−84 7.8612x10−8 7.6624x10−9 -2.2935x10−7

y120t 7.1008x10−8 7.6624x10−9 3.6238x10−8 -2.6791x10−7

πSt 7.9082x10−6 -2.2935x10−7 -2.6791x10−7 4.4052x10−4

d) Cholesky Matrix, P λt y1t y120

t πSt

λt 3.2912x10−3 0 0 0y1t 2.7848x10−5 2.7899x10−4 0 0y120t 2.1575x10−5 2.5311x10−5 1.8743x10−4 0πSt 2.4028x10−3 -1.0619x10−3 -1.5626x10−3 2.0765x10−2

VAR model found in Table 3.1, scenarios with long periods of contribution holidays and withunreasonably large salary increases was occurring frequently enough that any comparisonwould be doubtful. Also, a number of scenarios would require contributions so large (asmuch as 80% of the salary) that the plan would not be allowed to continue without seriousmodifications. Any comparison based on conditions that would not be allowed to exist wouldbe meaningless. Therefore, the VAR model was calibrated such that the extreme conditionsdescribed above would be rare enough not to bias our comparison of the two plans. By trialand error, it was determined that reducing the volatility in equity returns by about 50%would generate acceptable economic scenarios for the entire horizon considered. In orderto maintain the same contemporaneous correlation between variables, the covariance termsinvolving equity are divided by 2 and the variance of the equity return is divided by 4. Theresulting variance-covariance matrix, Σ, and its Cholesky decomposition, P , are given inTable 3.2. Together with the estimated matrix Φ in panel (b) of Table 3.1, we will refer tothis model as the calibrated one.

3.5 Simulated Results

Future economic scenarios are generated for a period of 60 years using the calibratedVAR model presented in the previous section. Let the innovation vector at future time hmonths from now, in a given scenario, be denoted by εh. Each vector consists of four inde-pendent Standard Normal variates. A total of n = 10, 000 such vectors are generated for720 months, h = 1,2, ... ,720.

14

The mean-adjusted economic variables, zh, h = 1,2, ... ,720, are obtained as follows:

zh = Φhz0 +h−1∑i=0

ΦiPεh−i. (3.10)

The initial value, z0, is the mean-adjusted vector of the last observed values for infla-tion, short-term interest rate, long-term interest rate and equity return. In April 2016, thevector of economic variables was x0 = (0.003122, 0.000416, 0.001162, 0.035727) and themean-adjusted vector was z0 = (0.001643, -0.002149, -0.002864, 0.029143).

Projected monthly rates, xh = (λh, y1h, y

120h , πSh )T , are obtained by simply adding the

mean vector, µ. That is

xh = zh + µ, (3.11)

where µ consists of the values found in panel a) of Table 3.1.

The economic variables that will be used in the next chapters are the effective rates perannum at an annual frequency. Therefore, the instantaneous monthly values, x, must firstbe converted into appropriate effective annual rates.

The instantaneous annual rate of inflation for year t, λt, is simply the compounded effectof the monthly rates of inflation during that year. That is,

λt =12t∑

h=12(t−1)+1λh, (3.12)

for t = 1, 2, ..., 60. Finally, the corresponding annual effective rate of inflation for year t is:

rinft = eλt − 1. (3.13)

Similarly, the instantaneous and annual effective equity returns in year t, t = 1, 2, ..., 60,are

πSt =12t∑

i=12(t−1)+1πSi (3.14)

15

and

rSt = eπSt − 1 (3.15)

respectively.

The returns on the fixed-income portfolio are obtained differently. We assume that theassets invested in short-term instruments pay a rate consistent with 1-month Treasury bills,in any given month, and that they are reinvested in a similarly way at the beginning of eachmonth. The annual effective rate of interest earned in year t on those assets, i12

t , is given by

i12t = exp

12t∑h=12(t−1)+1

y1h

− 1. (3.16)

Assets invested in long-term instruments are assumed to earn a rate consistent with theyield on 10-year bonds prevailing at the beginning of the year. The annual effective rate ofinterest on long-term investments throughout year t, i120

t , is therefore equal to

i120t = exp

(12y120

12t

)− 1 (3.17)

for t = 1, 2, ..., 60.

The fixed-income portfolio is invested 20% in short term instruments and 80% in long-term ones and is rebalanced at the beginning of each year. The annual effective return inyear t earned on that portfolio is:

rBt = 0.2i12t + 0.8i120

t . (3.18)

Figures 3.2 and 3.3 illustrate the expected values and standard deviations, respectively,of the simulated annual effective rates that will be used in the next chapters to compareDB and DC plans. They confirm that equity is the most volatile economic variable withthe highest expected value. Since the volatility of the equity returns was lowered in thecalibrated model, the standard deviation of the equity return in our projections is lowerthan the one observed in the historical data. The expected value and standard deviation forthe two bonds, 1-month and 10-year, have similar patterns; with the 10-year bond offeringa higher expected return. Those two series, on average, revert to their long term expectedlevels in about 20 years from now. It is also worth noting that the simulations start in alow interest rate environment.

16

Figure 3.2: Average of 10,000 annual rates generated by the calibrated model

The four economic variables are simulated at a monthly frequency for 60 years using (3.10) and (3.11) withthe covariance matrix found in panel b) of Table 3.1 and the Cholesky matrix in panel d) of Table 3.2. Eachof the 10,000 paths starts with the observed monthly rates in April 2016 for the economic variables, x0 =(0.003122, 0.000416, 0.001162, 0.035727). Corresponding annual rates are calculated using (3.12)-(3.17).

17

Figure 3.3: Standard deviation of 10,000 annual rates generated by the calibrated model

The four economic variables are simulated at a monthly frequency for 60 years using (3.10) and (3.11) withthe covariance matrix found in panel b) of Table 3.1 and the Cholesky matrix in panel d) of Table 3.2. Eachof the 10,000 paths starts with the observed monthly rates in April 2016 for the economic variables, x0 =(0.003122, 0.000416, 0.001162, 0.035727). Corresponding annual rates are calculated using (3.12)-(3.17).

18

Chapter 4

Defined Benefit Plan

4.1 Definition

A DB plan is a pension plan that promises its members a retirement income based onretirement age, years of service, salary and possibly other factors like, for example, entitle-ment to social security benefits. The retirement income may be partially or fully indexedto inflation. They come in many forms. In fact, the Pension Benefits Standards Act (1985)defines a DB plan as “a pension plan that is not a defined contribution plan”. An analysisof the range of DB plan designs is beyond the scope of this project.

Put simply, a typical DB plan guarantees its members some retirement benefits, calcu-lated by a pre-determined formula, which are funded through contributions generally madeby the employer and investment income on the pension funds. The level of contributionsvaries depending on the investment performance and the value of the actuarial liabilitiesrelative to the fund balance. Since the employer pays the contributions, it is a well-knownfact in the industry and the literature on DB plans that it is the employer who bears thisinvestment risk. The employer also bears other risks like the longevity risk, a risk not con-sidered in this project.

4.2 Investment risk

This investment risk is usually studied for its impact on the costs to the employer andon the funding ratio of the plan. Although the employer is the one directly and perhapsmostly impacted by the variable contributions, the plan members might also be impactedthrough lower future salaries, and to some extent through lower retirement income (sincethe benefit is a function of the salary earned).

19

In some, if not most situations, an employer who is required to made larger contributionsto a DB plan will have less room in its budget to cover the salaries of its employees andmake capital investments. This is certainly the case in some public sectors where the budgetof the employer is likely to increase at a rate linked to inflation. There are also examples ofthis in the private sector, where large pension deficits and financial hardships (not unlikelyto arise in the same periods) have forced employers to cut jobs, reduce salaries and pensionbenefits. One can find many examples of DB plan members who saw their pension benefitsreduced. For example, a number of employees in the automobile industry and the aerospaceindustry, in municipalities, in universities lived through periods of salary freeze (some hav-ing to settle for salary cuts) over the years. Even retirees for these employers were affectedin some cases, being forced to accept lower benefits than promised at retirement.

Although the benefits in a DB plan are generally considered guaranteed, this is not al-ways the case. They are guaranteed as long as the employer has the financial means to payany deficits (over a period of time) and make the required ongoing contributions. Stelco,Nortel and recently Sears are examples of bankruptcies where employees lost part, if notmost, of their pension benefits.

Our goal is to study how much of the investment risk is borne by the members of DBplan and compare it to the investment risk that a DC plan member would bear in the sameeconomic environment. For convenience, we will consider that the employer sponsoring therepresentative DB plan described in the next section operates in a simple controlled financialenvironment. The analysis will focus on the salaries and the defined benefits that DB planmembers will get under stochastically generated future economic scenarios. The risks thatthese members would see their retirement benefits reduced because of financial hardship ofthe employer, a plan termination or bankruptcy are ignored. Hence, the pension determinedat time of retirement is indeed guaranteed in our model.

4.3 Representative DB plan

The DB plan selected for our analysis is a 5-year final average salary. The benefit pro-vided is 2% per year of service of the average salary over the last 5 years of employment.This plan would provide a replacement ratio of approximately 70% to a member retiringafter 35 years of service. A 70% replacement ratio, i.e. the ratio of retirement income tofinal salary, is generally considered satisfactory to maintain one’s standard of living afterretirement (see for example, Aitken (1996)).

20

4.3.1 Assumptions

This section lists the key demographic and economic assumptions made in our model.The valuation assumptions are presented in Section 4.4.1.

The demographic assumptions are:1. All employees enter the plan when they start working for the employer at age 30.

2. The employer hires 100 new employees at the start of every year, all aged 30.

3. Retirement is mandatory at age 65 with no early retirement allowed.

4. There are no decrements while active, i.e. until retirement at age 65.

5. Retirees experience mortality rates based on the CPM2014 Public Sector MortalityTable. The mortality rates are unisex rates arbitrarily calculated as a weighted averageof 40% of male’s mortality rates and 60% of female’s rates at each year of age. Thelimiting age is set at 116.

6. The DB plan membership is stationary from time 0. It consists of 100 active employeesat each age between 30 and 65, and a distribution of retirees consistent with theassumed mortality table.

The main economic assumptions are:

7. The investment returns on the plan assets are net of any administration fees andmanagement expenses.

8. All assets are invested conservatively, 60% in equity and 40% in fixed-income withrebalancing happening annually.

9. Contributions are made and benefits are paid at the beginning of each year.

10. The starting salary of newly hired employees at time 0 is S0 = 65000.

11. The salary scale in any given year is based on a merit increase, m, of 1% per year ofservice.

12. Annual salary increases are negotiated every 3 years coinciding with the plan valua-tion. The rate is level and guaranteed until the next negotiation.

13. The operating budget of the employer increases annually at a rate equal to the rateof inflation experienced in the previous year.

14. The employer is not allowed to keep carrying forward a surplus or deficit. Therefore,any operating surplus experienced by the end of a 3-year cycle must be amortizedfully over the next 3-year cycle.

21

4.3.2 Notation

The notation used in this section is summarized below.• e: the entry age, which is 30.

• r: the retirement age, which is 65.

• nx,t: the number of members aged x at time t.

• ast : the annual salary rate of increase at time t. It is the rate used to adjust eachemployee’s salary at the beginning of year t. This rate is generally positive but canbe negative in some cases.

• Sx,t: the annual salary paid to an active employee aged x at time t. Note that Se,0 =S0 = 65, 000.

• TSt: the salary mass in year t. The sum of all salaries paid to active employees in yeart.

• FAEx,t: the 5-year final average earnings for a member aged x at time t.

• bx,t: the annual retirement benefit for a member aged x at time t, actual benefit forretirees, projected for active employees.

• Tbt: the total benefits paid to all retirees at time t.

4.3.3 Salaries

The salary paid to a member in a given year is his/her salary earned the previous yearadjusted by the total annual salary rate of increase, ast. This salary rate of increase includesthe merit increase, m, which is fixed at 1%, and the adjustment to the overall salary scalewhich depends on the financial situation of the employer and the DB plan. The determina-tion of this salary rate of increase, which is done every three years, is discussed in Section 4.5.

The salary of an employee aged x at time t can be obtained recursively. Starting withthe salary scale at time 0, which is by assumption,

Sx,0 =

S0 × (1 +m)(x−e) e 6 x 6 r − 1

0 otherwise,(4.1)

22

we obtain all future salary scales, for t = 1, 2, . . ., as follows:

Sx,t =

Se,t−1 × (1+ast)

(1+m) x = e

Sx−1,t−1 × (1 + ast) e < x 6 r − 1

0 otherwise.

(4.2)

The salary of new entrant at time t can also be calculated as Se,t = Se+1,t

1+m .

Past salary scales are needed to determine the benefits of those currently retired attime 0 and those retiring in the next 5 years. Here, we assume that the salary scale hasbeen increasing annually at a rate of 2%. This rate is equal to the Bank of Canada’s targetinflation rate.

So, for t = 1, 2, ...,

Sx,−t =

Sx,0 × (1.02)−t e 6 x 6 r − 1

0 otherwise(4.3)

The total salaries paid to all active members at time t is

TSt =r−1∑x=e

nx,t × Sx,t, (4.4)

4.3.4 Benefits

The 5-year Final Average Earnings for a retiree aged x, x > r, at time t is given bythe average of the 5 appropriate annual salaries calculated with the expressions above. Forexample, the FAE of a member retiring at age 65 at time 0, is

FAE65,0 = 15

5∑i=1

S65−i,−i. (4.5)

In general, for x > 65 and t = 0, 1, . . ., we have

FAEx,t = 15

5∑i=1

Sr−i,t−(x−r+i). (4.6)

The retirement benefits paid to the retirees of our representative DB plan are:

bx,t = 2% × (r − e) × FAEx,t, (4.7)

23

for x > r and t = 0, 1, 2, . . .

The total amount of retirement benefits paid to retirees at time t is

Tbt =115∑x=65

nx,t × bx,t. (4.8)

4.4 Actuarial Valuation

The employer of this DB plan is required to have an actuarial valuation done everythree years. In this section, the assumptions that will be made to determine the financialsituation of the plan are discussed. The funding method and the rules regarding the amor-tization of any actuarial surplus or deficit are presented. Each valuation will determine thecontributions that the employer will have to make to the pension fund for the next threeyears.

The Entry Age Normal (EAN) funding method, a commonly used actuarial fundingmethod will be adopted here. Under this method the value of the projected benefits for anemployee is allocated as a level percentage of salary between entry age and retirement age.The total annual contribution has two components, the normal cost for the year and theamortization of unfunded actuarial liabilities.

4.4.1 Valuation Assumptions

Valuing the actuarial liabilities of the plan requires a number of assumptions about thefuture. These assumptions can be classified into demographic ones and economic ones. Theyare commonly set at conservative levels compared to the expected experience of the plan.

For this project, the demographic valuation assumptions are exactly the same as theassumed experience. There is no conservatism built into these assumptions. For example,there is no decrement until the mandatory retirement age of 65 and retirees’ survival isbased on the CPM2014 Public Sector Mortality Table as described in Section 4.1.1.

The valuation of the plan liabilities will be a function of two key economic assumptions.The first one concerns future salary increases. Each year, employees get the merit increase,m = 1%, as well as an adjustment for inflation which is assumed to be 2%. So, for valuationpurposes, salaries increase at a rate of s = 3.02%, i.e. s = (1.01)(1.02)-1), at the beginning

24

of each year. The second economic assumption is the selection of the valuation rate. Thefuture cash flows, contributions and benefits, will be discounted at a rate consistent withthe expected future market returns at the time of the valuation, less a small spread to valuethe plan somewhat conservatively.

When determining the future expected market return at a given valuation time, theequity return will be set at its long-term average. For simplicity, it will be calculated as theaverage of the equity returns for the given year over the 10,000 scenarios. This average rateof return is very stable.

As for the return on the fixed-income investments, the assumption is that the rate willstart at the last observed return on fixed-income and will revert back towards some long-term average assumed to be 5%. For valuation purposes, fixed-income assets are assumedto earn the average of the current value and the 5% reversion level. This rate is path-dependent and may fluctuate significantly within and between scenarios. Specifically, thevaluation rate, ivt, is obtained as follows:

ivt = 0.6 × rSt + 0.4 ×[(0.5)(rBt ) + (0.5)(0.05)

]− sp, (4.9)

where rSt is the average of the 10,000 simulated equity returns for year t, and sp is thevaluation spread. Different values of spread ranging from 0 to 2% were tried in order tochoose one that produced stable results over the projection period of 60 years. A 50 basispoints spread was chosen, so sp = 0.005.

4.4.2 Actuarial Liabilities

The actuarial liability at time t is the value of assets that the plan should have accu-mulated to cover the accrued benefits of the plan members. This value is determined usingthe valuation assumptions in effect at time t.

Using the prospective method, the actuarial liability at time t, ALt, is the differencebetween the present value of all projected future benefits, TPVFBt, and the present valueof the projected future normal contributions, TPVFNC t. Mathematically,

ALt = TPVFBt − TPVFNCt. (4.10)

25

The present value of future benefits at time t in respect of a member aged x is

PVFBx,t =

bx,t×ar

(1+ivt)r−x 30 6 x 6 64,

bx,t × ax 65 6 x 6 115,

0 otherwise.

(4.11)

where the annuity factors, ar and ax are valued at the valuation rate, ivt, using the mor-tality table for retirees. Recall that the retirement age is r = 65.

An n-year life annuity due valued at an interest rate of i is calculated

ax:n =n−1∑t=0

tpx

(1 + i)t(4.12)

where tpx is the probability that someone age x will survive to age x + t. The value of awhole life annuity due on a life aged x, ax, is obtained by using the above expression withn = 115 − x.

Finally, the total present value of future benefits for all members is simply the sum ofthe PVFB for both actives and retired members. That is,

TPVFBt =115∑x=30

nx,t × PVFBx,t. (4.13)

4.4.3 Normal Cost

Under the EAN method, the Normal Cost (NC) is a level percentage of salaries con-tributed to the plan in respect of benefits accruing during the year. The level percentageof salaries, Ut, is calculated such that the present value of future benefits is equal to thepresent value of normal costs for a new entrant at the time of the valuation. The normalcosts for all active employees is that same percentage of their salaries.

The level percentage Ut is calculated as follows:

Ut = be,t × (1 + ivt)r−e × arSe,t × as

r−e. (4.14)

The annuity in the denominator of the above expression, asr−e , is one where the payment

increases at the assumed rate of salary increase s at the beginning of each year, and being

26

discounted at the current valuation rate, ivt. That is

asr−e =r−e−1∑i=0

(1 + s)i

(1 + ivt)i. (4.15)

The Normal Cost for an active employee aged x at time t is given by

NCx,t = Ut × Sx,t. (4.16)

The total Normal Cost for the DB plan in year t is the sum of the Normal Costs for allactive workers, which is

TNCt =64∑

x=30nx,t ×NCx,t. (4.17)

Having determined the Normal Cost for an employee aged x at time t, NCx,t, we canobtain the present value of future Normal Costs for that employee as follows:

PVFNCx,t = NCx,t × asr−x . (4.18)

Finally, since the Normal Cost is a level percentage of salary assumed to increase annuallyat a rate of s, the total present value of Normal Costs at time t is given by

TPVFNCt =64∑

x=30PVFNCx,t × nx,t. (4.19)

With the calculated values for TPVFBt and TPVFNC t, the actuarial liability, ALt, isknown from Equation 4.10.

4.4.4 Pension Fund

The contributions to the DB plan are pooled into a fund and the benefits to the retireesare paid out of the fund. The assets available at the beginning of the year are invested 60%into equity and 40% into fixed-income. The effective annual rate of return earned on thoseassets in year t, for t = 1, 2, . . ., is:

rmt = 0.6rSt + 0.4rBt , (4.20)

where the annual effective rates rSt and rBt are the returns on equity and fixed-income in-vestments respectively. They are defined in Chapter 3 and generated using the calibratedVAR model.

27

We will call the rate rmt the market return for year t since it represents the rate actuallyearned by investing the plan’s assets in the market.

Denoting the total contributions made to the fund at time t by Ct, the fund evolvesaccording to the following recursive equation for t = 1, 2, . . . :

Ft = (Ft−1 + Ct−1 − Tbt−1) × (1 + rmt−1). (4.21)

We will assume that the plan is fully funded at time 0. Otherwise the plan would startwith a surplus or a deficit which would affect the comparison with the DC plan. So, theinitial fund is

F0 = AL0. (4.22)

4.4.5 Unfunded Actuarial Liability

The plan’s experience over any three-year period (between valuations) is unlikely tomatch the assumptions made when calculating the Normal Cost. For example, the marketreturn will likely be different than the valuation rate. As a result, the accumulated pensionfund will not remain equal to the actuarial liability after time 0. The difference between theactuarial liability and the fund value on a valuation date is called the unfunded actuarialliability (UAL). We have, for t = 1, 2, . . .,

UALt = ALt − Ft. (4.23)

This UAL represents a surplus or a deficit which must be amortized over a period oftime. Different rules or regulations exist regarding the amortization of the UAL. We willuse a 15-year period to amortize any actuarial surplus or deficit that exists on any valuationdate. The annual amount needed to amortize the UAL over 15 years is a special contributionthat will be made to the pension fund each year for the 3 years following a valuation. Atthe end of those 3 years, a new UAL will be calculated and a new special contribution willbe determined. This cycle will be repeated every 3 years.

Denoting by APt the amortization payment determined at valuation time t, a specialcontribution made to the plan in years t, t+ 1 and t+ 2, we have

APt+s =

UALt

a15 ivt

s = 0; t = 0, 3, 6, . . .

APt s = 1, 2; t = 0, 3, 6, . . . ,(4.24)

where the 15-year annuity is valued at the valuation rate prevailing at time t, ivt.

28

Since the total contributions made to the fund at time t are

Ct = TNCt +APt, (4.25)

the recursive formula (4.22) for the pension fund can be written as:

Ft = (Ft−1 + TNCt−1 +APt−1 − Tbt−1) × (1 + rmt−1). (4.26)

Note that since the unfunded actuarial liability can represent a surplus or a deficit, thespecial contribution, AP , can be positive or negative; and can even make the total contri-bution, C, negative.

4.5 Business Setup

The business setup is such that the employer will have a limited budget to cover allsalaries of active members, pay the Normal Costs and make any special payments to theDB fund required to amortize the UAL. The plan contributions were discussed earlier inthis chapter.

The employer will negotiate salary increases with its employees every 3 years. A levelannual rate of salary increase will be guaranteed for those 3 years. In doing so, the employerwill take into consideration its expected budget over the next 3 years, the newly calculatedcontribution rate for the plan and the existing operating surplus or deficit since the lastvaluation. The main reason an operating surplus or deficit arises is that the actual annualbudget increases are linked to inflation which is not know at the time the guaranteed salaryincreases are negotiated. Finally, the employer is required to balance its budget and expen-ditures over each 3-year cycle.

The operating budget of the employer is adjusted for inflation at the beginning of eachyear. Our projections start with a balanced budget, no existing operating surplus or deficitat time 0 and as mentioned earlier no unfunded actuarial liability. The budget at time t isdenoted by Bt and given by

Bt = B0 ×t∏i=1

(1 + rinfi ) (4.27)

for t = 1, 2, . . .

29

Whenever the budget needs to be projected into the future, either implicitly or explic-itly, it will be done by increasing the last known actual budget at the target rate of inflationof 2% per annum.

4.5.1 Expenses

The available budget is intended to cover the expenditures, namely all salaries, total plancontributions and any amounts needed to eliminate any outstanding operating surplus ordeficit. These expenditures, or simply expenses at time t, Et, are equal to the sum:

Et = TSt + Ct + AOpDt (4.28)

where TSt is the total salaries defined in (4.4), Ct is the total contributions defined in (4.28)and AOpDt is the amount needed to amortize the operating surplus or deficit over 3 years.If there is an existing operating deficit, the employer is assumed to borrow money to coverit temporarily and required to repay it; the amount AOpDt will be positive. If there is asurplus, the employer will be allowed to spend it over the next 3 years and AOpDt will benegative. By assumption, we have no surplus or deficit at time 0, so AOpD0 = 0.

As mentioned above, the operating surplus or deficit arises when the expenses do notmatch the actual budget. The operating deficit at a valuation time t will be the sum of theannual deficits for the previous 3 years. That is, for t = 3, 6, . . .,

OpDt =3∑i=1

[TSt−i + TNCt−i +APt−i + AOpDt−3 −Bt−i]. (4.29)

Any operating surplus or deficit will be amortized over 3 years starting immediatelywhich gives

AOpDt+s =

OpDt

a3 rmt−1

s = 0; t = 0, 3, 6, . . .

AOpDt s = 1, 2; t = 0, 3, 6, . . . .(4.30)

The discount rate used in the 3-year annuity is the current market rate which meansthat the employer borrows (or invests) money at the rate last earned on the assets of thepension fund.

30

4.5.2 Salary Increase

In this business setup, the employer attempts to balance its budget by offering salaryincreases that can be afforded based on projections made at time t. We will assume thatthe actuarial valuation of the DB plan has been completed when salary increases are be-ing negotiated (or, shall we say, determined). Similarly, the operating surplus or deficitfor the 3-year period that just ended is known. And consequently, its impact on the bud-get for the next 3 years is known. The budget for the coming year, however, will only beknown at the start of that year and is therefore not used in determining the salary increases.

At the beginning of year t, the employer starts by assuming that its annual budget foreach of the next 3 years, i = 0, 1, 2, . . ., will be

Bt+i = Bt−1 × (1 + 0.02)i+1 (4.31)

where Bt−1 is the known budget for last year, B is a projected budget using a 2% inflationrate.

In an ideal world where no UAL and no operating deficit would ever occur, an em-ployer should be able to increase its salary mass by 2% each year. Each employee, exceptnewly hired ones, would get a salary increase of 3.02%, including the 1% merit increase.Unfortunately, it is unlikely that experience will match all assumptions, generating UALand operating deficits. So, salary increases will have to be set at levels different than theexpected 3.02%. For example, consider an employer with an operating deficit at the end ofa 3-year period. To balance its budget again, salaries should increase at a rate lower than3.02%. This will tend to lower the actuarial liability (at least compared to its value hadthe salaries being increased by 3.02%) and produce an actuarial surplus. So next valuation,chances are that the employer would be able to afford to increase salaries by more than3.02%. This would tend to increase the actuarial liability at the following valuation andgenerate an actuarial deficit. The next salary increase would have to be lower than the tar-get. The pattern would repeat itself and produce unstable funding ratios and very volatilesalary increases. This approach was tried and indeed produced unstable results.

To help smooth the results, the employer will consider projected results over a slightlylonger term. When determining the rate of salary increase at valuation time t, the employerwill aim for a projected balance budget in 4 years, i.e. in year t + 3. Let ∆t+3(ns) be thedifference between the projected budget and the projected expenses in year t+ 3 assumingannual salary increases at a new negotiated rate ns. This difference is the projected operating

31

surplus for year t+ 3 and is given by

∆t+3(ns) = Bt+3 − Et+3(ns) (4.32)

where Et+3(ns) are the projected annual expenses for year t+ 3 assuming salary increasesof ns per annum, calculated using (4.37) with corresponding projected components.

The goal is to find the value of ns that makes this projected operating surplus equal to0. We obtain a value for ns by linear interpolation between two carefully chosen rates, ns1

and ns2. Starting with ns0 = .0302, we calculate nsi, i = 1, 2, as follows:

nsi = nsi−1 + 0.00058 × ∆t+3(nsi−1)1, 000, 000 × 1.02t . (4.33)

The factor 0.00058 in the above formula is approximately the change in the rate of salaryincrease necessary to eliminate a projected surplus of 1, 000, 000 at the end of the first 3-yearcycle. Because the projected surplus or deficit is very sensitive to small changes in the rateof salary increase and with a non-linear relationship, it is important to select values of nscarefully. Since the impact of a salary increase on the projected surplus is proportional tothe salary mass, the adjustment factor needs to reflect inflation. Hence, the 1.02t factor thatappears in the denominator.

The annual rate of salary increase awarded to all employees in years t to (t+2) is definedas

ns = ns2 − ∆t+3(ns2) × (ns2 − ns1)∆t+3(ns2) − ∆t+3(ns1) . (4.34)

This rate, ns, becomes the rate awarded in years t, t + 1 and t + 2 (i.e. as in (4.2) forthose years).

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4.6 A Deterministic Scenario

This section illustrates how the salary increases are determined. Using a deterministicscenario based on the expectations for future economic variables, results are shown at threekey stages leading to the determination of the salary increase for years 3, 4 and 5.

In Table 4.1, the assumed business setup is presented. At time 0, a newly hired employeeearns a salary of 65,000. The total rate of salary increase, for merit and inflation, is 3.02%for the next two years.

Using the initial valuation rate, the total present value of benefits and contributions arecalculated. It is determined that the plan has a 1.925 billion actuarial liability. Since theplan is assumed to be fully funded, the value of F0 is also 1.925 billion.

The expenses for the first year are the sum of the salaries, TS0, and all contributions,TNC0. The budget is balanced in the first year, so B0 is equal to the expenses (309,476,205).

Table 4.2 shows the results known to the employer immediately after the actuarial valua-tion performed at the end of the third year, i.e. just before t = 3. At that time, the valuationrate is up to 6.12%. With the higher salaries, the actuarial liabilities are up to 2.043 billions.The pension fund received annual contributions, paid the benefits to retirees and earn themarket return allowing it to grow to 2.037 billions. The plan has an unfunded actuarial lia-bility of 6,622,291 was will have to be amortized over 15 annual special payments of 640,094.

Also, the employer ends up with an operating deficit because its budget increased atinflation which was less than the assumed 2% included in the salary increases. To eliminatethis operating deficit, the employer must pay 799,167 in each of the next three years.

Table 4.3 shows the results of the budget planning done by the employer and leadingto the negotiated salary increases for the next three years. The actual budget at time 2 isassumed to grow at rate of 2% per year. As explained in the previous section, the goal is toset the salary increase such that the budget will be balanced starting at time 6, that is afteramortizing the UAL and eliminating the operating deficit realized at time 3. This is achievedby solving for the salary increase that will make the projected deficit, B6 − E6, equal to 0.Since most values beyond time 2 in this table depend on the salary increase awarded in thenext three years, there is no explicit solution. The linear interpolation method describedin the previous section is used to approximate the desired salary increase. Awarding salaryincreases of 3.2% will result in a projected deficit at time 6 of 4.16 which is close to 0.

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Table 4.1: Business setup assumed at time 0, DB plan under a deterministic scenario

Time, t 0 1 2 3 4 5 6

Starting Salary, Se,t 65,000 66,300 67,626

Salary Increase, ast 0 .0302 .0302

Salary Mass, TSt 270,791,791 276,207,627 281,731,780

ivt 0.056757

TPVFB0 2,481,573,831

Normal Cost rate, U0 0.1429

TNC t 38,684,413 39,458,102 40,247,264

TPVFNC 0 507,545,906

AL0 1,974,027,925

F0 1,974,027,925

UAL0 0

AP0 0

Budget, Bt 309,476,205

Expenditures, Et 309,476,205

OpD0 0

AOpD0 0

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Table 4.2: Information known after valuation at time 3, DB plan under a deterministicscenario

Time, t 0 1 2 3 4 5 6

Starting Salary, Se,t 65,000 66,300 67,626

Salary Increase, ast 0 .0302 .0302

Salary Mass, TSt 270,791,791 276,207,627 281,731,780

ivt 0.056757 0.058869

TPVFBt 2,481,573,831 2,558,119,837

Normal Cost rate, Ut 0.1429 0.1429 0.1429 0.1347

TNC t 38,684,413 39,458,102 40,247,264

TPVFNC t 507,545,906 498,344,561

ALt 1,974,027,925 2,094,854,227

Ft 1,974,027,925 2,088,080,689

UALt 0 6,773,538

APt 0 646,052 646,052 646,052

Budget, Bt 309,476,205 314,583,590 320,780,160 327,195,765

Expenditures, Et 309,476,205 327,511,512

OpDt 0 2,281,020

AOpDt 0 803,888 803,888 803,888

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Table 4.3: Projections used to determine salary increase, ns, at time 3, DB plan under adeterministic scenario

Time, t 0 1 2 3 4 5 6

Starting Salary, Se,t 65,000 66,300 67,626 69,152 70,713 72,309

Salary Increase, ast 0 0.0302 0.0302 0.032795 0.032795 0.032795 0.0302

Salary Mass, TSt 270,791,791 276,207,627 281,731,780 288,090,237 294,592,200 301,240,907 307,265,725

ivt 0.056757 0.058869 0.058869

TPVFBt 2,481,573,831 2,557,303,136 2,715,574,901

Normal Cost rate, Ut 0.1429 0.1429 0.1429 0.1347 0.1347 0.1347 0.1347

TNC t 38,684,413 39,458,102 40,247,264 38,805,324 39,681,128 40,576,699 41,374,687

TPVFNC t 507,545,906 498,344,561 532,853,510

ALt 1,974,027,925 2,094,854,227 2,182,663,197

Ft 1,925,965,816 2,088,080,689 2,220,650,992

UALt 0 6,773,538 -37,987,795

APt 0 646,052 646,052 646,052 -3,666,632

Budget, Bt 309,476,205 312,066,074 318,213,056 327,195,765 333,739,680 340,414,474 347,222,763

Expenditures, Et 309,476,205 328,339,536 347,185,885

Bt − Et 0 6.80852

OpDt 0 2,281,020 6,313,357

AOpDt 0 803,888 803,888 803,888 2,230,902

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4.7 Projected Salary Increases and Contribution Rates

Figure 4.1 illustrates the range of salary increases that the employer might be able toafford based on our 10,000 stochastic economic scenarios. Starting at a value of 3.02%,the annual rate of salary increase occasionally goes up to 10% in some scenarios. On thedownside, the employer has to cut salaries by 4% in a number of scenarios. Those extremecases only happen in 1% of the best or worst scenarios generated. The long-term mean ofthe rates of salary increase stays around the 3% level which is the salary increase assumedfor the first 3 years of our projection period. The middle-quartile range indicates that mostsalary increases will fall between 1% and 5%. These results seem quite reasonable.

Figure 4.1: Projected Salary Increases for Sixty Years

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Figure 4.2 shows the Normal Cost rates over our projection horizon of 60 years. Thestarting Normal Cost percentage was calculated to be 13.4%. The Normal cost rate steadilydeclines to a level around 11.5% over the first 15 years and remains stable at that leveluntil the end of the projection horizon. The reduction in the Normal Cost percentage ispartly due to the choice of a conservative valuation rate, which enables the pension fund tocarry more assets than what would be just necessary to cover the expected costs of futurebenefits and partly due to the fact that we expect fixed-income returns to increase fromtheir current low levels towards higher long-term means.

Figure 4.2: Projected Normal Cost Rates for Sixty Years

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Chapter 5

Defined Contribution plan

5.1 Definition

A Defined Contribution (DC) plan is a pension plan where the contributions are of a setor defined amount. The contribution amount is often a fixed percentage of an employee’ssalary. Each plan member has an individual account in which contributions are depositedand investment earnings are credited. DC plans may also be called a Money Purchase Plans(MPP).

The retirement income of a member depends on the amount accumulated in his/herindividual account at the time of retirement. The member has a number of options at re-tirement ranging from continuing to manage the account and draw income from it to usingthe balance in the account to buy an annuity from an insurance company.

DB and DC pension plans face many of the same risks. The level and impact of a givenrisk may vary between the two types of plans. How the existing risks are shared betweenthe employer and the employees is arguably one of the most discussed topic in the pensionliterature. In the next section, we discuss briefly the most important risks involved with aDC plan.

5.2 Risks with a DC Plan

Unlike DB plan members, participants of a DC plan will enjoy stable contribution rates.However, the DC members will bear most of the risks associated with having an individ-ual account. With DB plans, the members essentially pool the risks and therefore sharethem with other members, past, present and future throughout built-in intergenerationaltransfers. DC members owning their individual accounts without the risk of having to cover

39

any deficits in the plan; and having the opportunity to manage their funds successfully areadvantages cherished by some employees.

5.2.1 Investment Risks

The member bears the investment risk during the accumulation period, i.e. while ac-tively employed and contributing to the plan. The member is usually offered a number ofoptions to invest the funds available in his/her account. Therefore, the member is bearingthe risk associated with market fluctuations in relation to his/her investment strategy. De-pending on the financial literacy of the member, this risk may be quite significant.

The DC member is also impacted by the timing of market volatility and market cor-rections. The volatility in the financial market is known to vary over time and marketcorrections tend to happen more frequently than some would like to admit (for example,in Figure 3.1 we can see large negative market returns in 2003, 2008 and more recently in2012). A member who sees his DC account depleted because of a market crash early inhis/her career will have time to recover with many years of future contributions and invest-ment earnings. The member who is near retirement when a market crash happens will notbe so lucky. The depleted account might force the member to retire with a low income orwork longer.

5.2.2 Interest Risk

A DC plan member who wishes to buy a life annuity at time of retirement, or laterduring retirement, will face the risk of having to do so when interest rates are low, andtherefore annuity prices high. This is the interest risk. This is not a risk borne by DBmembers, provided the plan remains solvent. In addition, a DC plan member interested inbuying a life annuity will generally not be able to do so at the lower group rate effectivelyoffered to DB participants (McCarthy 2003).

5.2.3 Other Risks

DC plan members who choose not to buy a life annuity will face the risk of outlivingtheir assets. This is the longevity risk. Compared to DB plan members, inflation will havea greater effect on DC members. The rate of inflation during employment and after retire-ment may affect DB and DC members differently. Some DB plans offer retirement incomesthat are partially indexed to inflation. These other risks are beyond the scope of this project.

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5.3 Representative DC Plan

5.3.1 SFU Pension Plan

SFU offered a non-contributory DB plan from 1964 to 1973. In March 1973, the planwas amended to a DC plan that provided a fully vested and portable individual account toexisting members. Plan members at the conversion date (called the Closed Group) are en-titled to the greater of a life annuity purchased with their individual account or the benefitdetermined by the old formula. The cost of the formula benefit was funded by a surcharge of2.24% for 2 years, then 2% for 5 years, of the basic salary of members in the Closed Group.The surcharge was paid by levies from the individual accounts of the DC plan members.

Some of the key features of the SFU Academic Pension Plan taken from the SFU 2016Annual Report of Academic Pension Plan (SFU, 2016) and the SFU Pension Plan for Aca-demic Staff Summary (SFU, 2013) are summarized below.

• ContributionsThe University makes a compulsory contribution of 10% of a member’s basic salaryless the University’s required contributions to the Canada Pension Plan (CPP). Eachplan member has an individual account called Money Purchase Account (MPA). Aplan member is permitted to make voluntary contributions to his/her MPA up to acertain percentage (18% in 2016) of his/her earnings less the amount of contributionmade by the employer. The portion of the MPA due to the employer’s contributionscannot be withdrawn until retirement. The member can withdraw any voluntary con-tributions, with the corresponding investment earnings, while employed at SFU.

• RetirementNormal retirement date is the first day of September following attainment of age 65.Early retirement is allowed on the first day of any month after attaining age 55. Re-tirees at normal retirement date have the option to purchase several different types ofannuities at rates quoted by Sun Life Financial or leave the funds in their MPA untilage 71.

• Investment choicesIndividuals have a number choices when deciding on an investment strategy. Theycan allocate assets in a range of investment funds, or choose one of the InvestmentFunds offered. The default fund is the Balanced Fund which has a target allocationof 65% equities and 35% fixed income. The Balance Fund earned annual returns of

41

6.8%, 6.3% and 11.5% in year of 2016, 2015 and 2014 respectively.

Members of the SFU DC plan are starting to retire in larger numbers after spendingtheir academic careers as SFU employees. Due to recent market performances and a low-interest rate environment, some members retiring are finding that the benefits provided bytheir MPA are inadequate. This is certainly a strong reminder that the members of a DCplan are the ones bearing the investment and interest risks. However, to conclude that DCplans are most expensive and riskier than DB plans on this basis is ignoring the possibleoutcomes for future generations of plan members. It should also be pointed out that oneshould expect to contribute more than 10% of salary to adequately fund his/her retirement.

In this project we undertake not only to compare the benefits provided by both plans ina wide range of economic situations but also to study the impact that given environmentsmay have on the salaries earned by members while employed. For our comparative analysis,a representative DC plan with characteristics similar to an existing one, the SFU AcademicPlan, was chosen.

5.3.2 Design and Assumptions

The SFU plan forms the basis for the selection of the representative DC plan used inthis project. We assume a fixed 10% contribution rate of basic salaries paid entirely by theemployer. For simplicity, there are no voluntary contributions from the members.

The representative member is investing the assets available in the MPA in a fund con-sisting of 60% equities and 40% fixed-income. The asset mix is rebalanced at the beginningof every year. This investment strategy is consistent with the one adopted by the plan spon-sor of the DB plan studied in Chapter 4.

The demographic assumptions, where appropriate, are the same as those described inSection 4.3.1.

5.4 Business Setup

For a fair comparison, the budget of the employer is exactly the same as the one for theemployer offering our representative DB plan. That is, the budget increases annually withinflation. The employer is not allowed to carry any surplus or deficit. Consequently, every

42

year the full budget is spent on salaries and DC contributions. As was the case with the em-ployer sponsoring the DB plan, there are no other expenses to be covered with this budget.This setup makes the projections of future salaries and DC accounts a fairly simple exercise.

Under the design of our representative DC plan, in any given year t, 110% of totalsalaries would use up the full budget for that year. The amount available for all salaries inyear t, TSDCt , is

TSDCt = Bt1.1 (5.1)

where Bt is the budget amount described in Section 4.5.1.

With 100 members at each age and a 1% merit increase per year of service, the totalsalaries in year t is also given by

TSDCt =r−1∑x=e

nx,tSDCx,t

= 10034∑i=0

SDCe,t (1 +m)i

= 100SDCe,t s35m

(5.2)

where SDCx,t is the salary earned by an employee aged x in year t.

The annuity certain is valued at a rate of m = 0.01 which gives s35m = (1+m)35−1m =

41.6603.

Substituting (5.2) into (5.1) and solving for the salary earned by a new entrant at t, weget

SDCe,t = Bt1.1s35m

. (5.3)

The annual salary of a member of the DC plan who is aged x at time t is

SDCx,t = SDCe,t (1 +m)x−e (5.4)

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and the amount contributed to the MPA of this member is

CDCx,t = 0.1 × SDCx,t . (5.5)

The contributions will accumulate in the MPA for 35 years and provide a lumpsum atthe time of retirement of the member. Denoting the fund available at time t+s in the MPAof a member who was hired at time t by FDCt,s , we have the following recursive equation:

FDCt,s+1 = [FDCt,s + CDCe+s,t+s] × (1 + rmt+s+1) (5.6)

where FDCt,0 = 0 for all t, that is new entrants start with no funds available in their MPA.

5.5 A Deterministic Scenario

Unlike the complex actuarial valuation required for a DB plan, a DC plan account issimple and straightforward to track. The numerical results for an individual account of ourrepresentative DC plan, under the same deterministic scenario that was used to illustratethe DB plan results in Section 4.6, up to year 6 are shown in Table 5.1. The table showsthe balance in the MPAs for the first two cohorts of new employees.

Again, the budget amounts are exactly same as those obtained when studying the DBplan. The salary of a newly hired employee in a given year is determined by Equation (5.3).The contribution amount which is 10% of the salary is deposited into the MPA at the be-ginning of the year. The fund balance earns the market return, rm, for that year.

Some of the key numbers in Table 5.1 are calculated as follows:

The starting salary at time 0 is SDCe,0 = 66, 992 = 306,999,561(1.1)(41.6603)(100) ,

The salary at time 1 of a new entrant at time 0 is SDCe+1,1 = 68, 779 = 66, 992(1.016503)(1.01),

The accumulated fund at time 1 for a new entrant at time 0 is FDC0,1 = 13, 956 =6699(1.056549)+6878 and the starting salary of a new entrant at time 1 is SDCe,1 = 68, 098 =66, 992(1.016503).

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Table 5.1: Individual Accounts for first 2 cohorts of employees in the DC plan under adeterministic scenario

Time, t 0 1 2 3 4 5 6

Inflation, rinft 0.016503 0.019698 0.018867 0.018734 0.018628 0.018539 0.018463Market return, rmt 0.054186 0.055734 0.057159 0.058411 0.59513 0.060484 0.061340

Budget, Bt 309,476,205 314,583,590 320,780,162 326,832,208 332,954,982 339,157,134 345,444,753

New entrant at 0:

Salary, SDCe+t,t 67,532 69333 71,406 73,481 75,606 77,784 80,019

Contribution, CDCe+t,t 6753 6933 7141 7348 7561 7778 8002

Fund balance, FDC0,t 6753 14,053 21,976 30,581 39,927 50,082 61,113

New entrant at 1:

Salary, SDCe+t−1,t 0 68,647 70,699 72,753 74,857 77,014 79,227

Contribution, CDCe+t−1,t 0 6865 7070 7275 7486 7701 7923

Fund balance, FDC1,t 0 6865 14,317 22,411 31,206 40,764 51,152

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Chapter 6

Comparison of DB and DC Plans

The SFU Faculty Association is considering the possibility of switching from the exist-ing DC plan to the B.C. College Pension Plan, which is a jointly sponsored DB plan. Areport prepared by PBI Actuarial Consultants Ltd (2015) compares the retirement incomefor existing members based on their current age and retirement age under a number ofdeterministic scenarios. It is suggested that the B.C. College Pension Plan would reduceinvestment risk, inflation risk and longevity risk. It concludes that in most scenarios, mostmembers would do better in the College Pension Plan than in their current DC plan. Thedeterministic scenarios and the analysis of retirement benefits only are two of the limitationsof the study. The motivation of this project is to use a stochastic model and take futuresalaries into account.

At the time of a pension plan conversion, existing members may be given the option tostay in the old plan. New employees, however, are required to join the new plan. The finan-cial situation of the old plan may make it more or less advantageous for existing membersto stay in it. For this reason, our comparison of the two plans will exclude existing members.

The focus of our comparison is on the first 25 cohorts to join the plans after time 0. Wewill project their experience with the employer and the pension plan until the last cohortreaches retirement age, after 35 years of service. Therefore, the projections will run for 60years.

We consider two employers operating in the same conditions with the same availablebudget to cover salaries and all contributions to the pension plan. The only difference is thetype of pension plan they sponsor, one offers a DB plan, the other a DC plan.

Employees join the plan offered by their employer at the time they get hired. Sincecontributions vary over time and differ between plans, the employees of the two employersearn different salaries. However, the salary scale is based on a 1% merit increase in both

46

cases. The new employees in the DB plan may join at a time when Normal Costs are lowand consequently, start at a relatively higher salary. Conversely, they may start when thereis a funding deficit requiring higher contributions and have to settle for a lower salary. Sofuture cohorts may be affected differently than the initial cohort hired at time 0.

In this chapter we analyze how well the cohorts in one group do compared to corre-sponding cohorts in the other group. The total compensation received by a cohort is themeasure of choice here to compare the two plans. Total compensation is arbitrarily definedas the value at time of retirement of all salaries earned during the 35-year career plus theequivalent lumpsum available in the pension plan at time of retirement. For this purpose,the salary accumulation is done at a rate that can be interpreted as a preference rate of theemployee for future cash flows. We consider two rates, the first is a low discount rate corre-sponding approximately to the average inflation over all the projections. A second discountrate of 6%, which is closer to the average market returns, is also considered. Other ratesand approaches, for example, using an expected utility function, can be used to measurethe value of the total compensation received by a plan member.

The comparison will focus on the difference between total compensations for membersof the two plans. These differences will be cohort-specific as later cohorts may be affecteddifferently than earlier ones. We will use the concept of Value-at-risk (VaR) to compare theplans. The graphs presented in this chapter will all show the mid-quartile range as well asfour quantiles, namely, 1%, 10%, 90% and 99%, of the 10,000 generated results. Althoughnot particularly relevant to one employee per se, means and medians are also shown in eachgraph.

Finally, the difference obtained between the two plans for the deterministic scenario willalso be included in each graph. We call the difference in the deterministic case the baseline.It is important to recognize that the two plans do not start exactly on the same basis. TheDB plan starts as a mature and fully funded plan with assets above the expected liabilityof the plan. Recall that the actuarial liability is valued at a conservative level. This excessfunding over the expected cost of the benefits was contributed in part by past cohorts whohave long left the employer. This represents an advantage to existing employees of the DBplan. The DC members start their own MPA from scratch when they get hired with no sub-sidy from other employees. The distribution of results obtained from the 10,000 simulationsare informative in themselves but also in relation to the baseline.

Before presenting the results for the total compensation, it is interesting to look at keyintermediary values and major components of the total compensation. It will provide a bet-ter understanding of where the differences in compensation come from. We will discuss the

47

distributions of the starting salaries of the 25 cohorts, the value of their retirement fundsat age 65 and their total compensations.

6.1 Comparison of Starting Salaries

Figure 6.1 shows key values of the distribution of the 10,000 simulated differences inthe starting salaries of the 25 cohorts. The DB plan members contribute about 13% of thesalaries at time 0 which is more than the 10% contributed by the DC plan members. Con-sequently, their salaries are less. After about 10 years, the DB Normal Cost rates remainfairly stable around 11.5%, which is still higher than the 10% contribution rate of the DCmembers. However the DB plan tends to generate a surplus each year, partly due to theincreasing trend in the valuation rate (recall that the projections start in a low interest rateenvironment). This plan surplus, in our model, is redistributed in the form of higher salarieswhich explains that starting salaries of DB employees on average exceed those of the DCemployees in later years.

Figure 6.1: Difference in starting salaries by cohort between DB and DC plans

48

6.2 Comparison of Retirement Benefits

The amount available in the individual account of a DC plan member is, in some sense,the present value of his/her retirement benefits. This lumpsum can be managed to drawincome or used to purchase a life annuity. A DB plan member does not have an accountbalance at retirement. The member instead has the right to receive a defined benefit for life,provided the plan remains funded. The value used for comparison will be the discountedvalue of his/her life contingent payments at a rate equal to the valuation rate prevailingat the time of retirement. The fair market value of the annuity would need to take intoaccount a number of factors not modeled here, like market rates used by insurance com-panies to price annuities, the risk of plan insolvency, the risk bankruptcy of the employer,the management fees associated with maintaining the DC account, etc. The plan valuationrate determined in each scenario at the time a member retires, is a reasonable rate that canbe used to approximate the present value of the benefits and study the volatility associatedwith the annuity payments.

Figure 6.2 presents the results of 10,000 simulations of differences between lumpsums ofthe two plans. We see that in more than 75% of the scenarios, the DB members will receivea higher value of retirement benefits.

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Figure 6.2: Difference in values of retirement benefits by cohort between DB and DC plans

6.3 Comparison of Total Salaries

Looking only at the value of the retirement benefits ignores the important component ofan employee that is the wages earned while actively employed. Providing adequate retire-ment benefits are not very appealing if the employee must work for a salary placing him/herunder poverty level for 35 years. Conversely, a plan providing lower retirement benefits maybe more than adequate overall if employees are given higher wages allowing them to makeadditional contributions to supplement their retirement income, for example in a registeredretirement savings plan (RRSP).

Figure 6.3 and Figure 6.4 show the difference in the value of all salaries received whileworking, by cohort, between the DB members and DC members. In Figure 6.3 the salariesare accumulated at 1.8%, the approximate average of the rate of inflation. Salaries are ac-cumulated at 6% in Figure 6.4.

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Figure 6.3: Difference in accumulated values of salaries at 1.8% by cohort between DB andDC plans

51

Figure 6.4: Difference in accumulated values of salaries at 6% by cohort between DB andDC plans

6.4 Comparison of Total Compensation

In the next two figures, the value of the retirement benefits are added to the accumulatedvalue of the salaries. Each sum is for a specific scenario and a particular cohort. Figure 6.5is for the case where salaries are accumulated at 1.8%. Figure 6.6 corresponds to salariesaccumulated at 6%.

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Figure 6.5: Difference in values of total compensation by cohort between DB and DC plans,1.8% case

53

Figure 6.6: Difference in values of total compensation by cohort between DB and DC plans,6% case

Based on the results shown in Figure 6.5 and Figure 6.6, our projections seem to indicatethat the DB members would be better compensated overall than DC members. This is trueat rates of 1.8% and 6%, perhaps suggesting that it would be the case at other preferencerates and possibly at whatever rate would be deemed a fair market rate or even when onewould consider expected utilities.

6.5 Comparison of Total Compensation Relative to Baseline

As mentioned earlier, the DB plan in our model is a fully funded plan at time 0, partly ormostly due to contributions in excess of the expected cost of benefits made by past cohortsof employees. Together with the fact that the valuation rate tends to increase over timein our projections, allows the plan to slowly release emerging surpluses to later cohorts ofemployees. In other words, the setup implicitly favors DB members. Had the projectionsstarted in a high interest rate environment, the conclusion might have been different. As anattempt to correct for this slight bias in favor of DB members, the baseline results (i.e. the

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deterministic case) are subtracted from the projected results. Differences in total compen-sation, baseline-adjusted, with salaries accumulated at 1.8% and 6% are displayed in Figure6.7 and Figure 6.8 respectively.

Figure 6.7: Baseline-adjusted difference in values of total compensation by cohort betweenDB and DC plans, 1.8% case

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Figure 6.8: Baseline-adjusted difference in values of total compensation by cohort betweenDB and DC plans, 6% case

The baseline-adjusted results suggest that employees of either plan are likely to receivemore or less equivalent total compensations. The differences in compensation are due tofuture economic environments, some favoring one type of plan over the other.

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Chapter 7

Conclusion

This project was inspired by the possible transfer of the SFU Faculty Association DCplan to a jointly sponsored DB plan. Pure DB and DC plans have been the subject of manystudies in the literature but their findings are far from showing some agreement as to whichplan is better.

Here, the framework used to compare these two types of plans, although simplistic, isnovel in the pension literature. The total compensations that the members of each typeof plan would received throughout their working careers and during their retirement yearsform the basis of the comparison. Stochastic economic scenarios are generated by a four-dimensional VAR model and used to project the future financial situation of the employerand its impact on the wages and retirement incomes of the employees. The employer hasa set budget, indexed to inflation, to be spent entirely on salaries and contributions to thepension plan. As a result, increasing the required pension contributions would adverselyimpact the current and/or future salaries of the employees.

A deterministic scenario corresponding to the average projections of the four choseneconomic variables serves as the baseline results. In this baseline case, DB members do bet-ter than the corresponding DC members. Two main reasons explain the difference. First,the DB plan is valued conservatively and is assumed to be fully funded at the start of ourprojections. Secondly, the current low interest rate environment which is expected to revertback towards higher long term levels tend to lower the actuarial liabilities of the DB plan,therefore releasing funds to the employer which are then slowly redistributed to employeesin higher salaries. Those are advantages not offered to DC plan members. Had the projec-tions started with an underfunded DB plan and/or in a high interest rate environment, theDB plan members would have been at a disadvantage.

To correct for the bias due to the starting conditions of our projections, the baselineresults are subtracted from the stochastically generated ones. With this correction, we find

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that the two types of plans offer similar total compensations to their members, with com-parable averages and distributions.

The simple setup in which the two types of plans are compared in this project can beextended in many ways. Future work could better reflect differences that exist in practice,some in favor of the DB plan, others in favor of the DC plan. We mention a few suggestionsfor future work.

The life annuity promised to the DB member was valued at a rate that would not beavailable to the DC member. One could project market prices of life annuities and use thoseprices when DC members choose to buy an annuity at retirement.

Investment performances were assumed to be identical between the two types of plans.This is unlikely to be the case in practice. A DB plan can negotiate lower management feesand can rely on financial experts to manage the assets of the plan. A DC member, on theother hand, may choose to invest more aggressively or adopt a life-cycle investment strategyto increase expected returns.

The risk aversion of a member could be taken into account when considering his/herwages and retirement incomes. Using a utility function would better reflect preferences ofdifferent cash flows over time .

Many DB plans are underfunded, a comparison of the two types of plans assuming dif-ferent initial funding ratios for the DB plan would be interesting.

Another possible extension, although more difficult to model, would be to allow theemployer more flexibility with regard to its budget. For example, the employer could lay offsome employees or delay hiring new ones when necessary.

Studying total compensations offered by different hybrid pension plan designs could alsobe the subject of future research.

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Appendix A

Mortality table used for retirees

Figure A.1: CPM2014 Public Sector Mortality Table

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